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In triangle abc AB = 6 ,BC = 5 ,CA= 4 and AP bisects angle A.if p lies on BC then BP equal to?
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In triangle abc AB = 6 ,BC = 5 ,CA= 4 and AP bisects angle A.if p lies...
Given:
- Triangle ABC with AB = 6, BC = 5, and CA = 4
- Angle AP bisects angle A
- Point P lies on BC
To find: BP
Solution:
Step 1: Find the length of AC
Using the Law of Cosines, we can find the length of AC:
AC² = AB² + BC² - 2(AB)(BC)cos(A)
AC² = 6² + 5² - 2(6)(5)cos(C)
AC² = 61 - 60cos(C)
AC² = 61 - 60(4/5)
AC² = 1
AC = 1
Step 2: Find the length of AP
Since AP bisects angle A, we can use the Angle Bisector Theorem:
AP/PC = AB/CB
AP/PC = 6/5
AP = (6/11)(PC)
Now we need to find PC. We can use the Law of Cosines again:
PC² = AC² + AP² - 2(AC)(AP)cos(C/2)
PC² = 1 + [(6/11)(PC)]² - 2(1)(6/11)(PC)(√[(1+cos(C))/2])
Solving for PC, we get:
PC = 55/26
Substituting this value into the equation for AP, we get:
AP = 30/143
Step 3: Find BP
Using the Angle Bisector Theorem again, we have:
BP/PC = AB/AC
BP/(55/26) = 6/1
BP = 330/55
BP = 6
Therefore, BP = 6.
Final Answer: BP = 6.
- Triangle ABC with AB = 6, BC = 5, and CA = 4
- Angle AP bisects angle A
- Point P lies on BC
To find: BP
Solution:
Step 1: Find the length of AC
Using the Law of Cosines, we can find the length of AC:
AC² = AB² + BC² - 2(AB)(BC)cos(A)
AC² = 6² + 5² - 2(6)(5)cos(C)
AC² = 61 - 60cos(C)
AC² = 61 - 60(4/5)
AC² = 1
AC = 1
Step 2: Find the length of AP
Since AP bisects angle A, we can use the Angle Bisector Theorem:
AP/PC = AB/CB
AP/PC = 6/5
AP = (6/11)(PC)
Now we need to find PC. We can use the Law of Cosines again:
PC² = AC² + AP² - 2(AC)(AP)cos(C/2)
PC² = 1 + [(6/11)(PC)]² - 2(1)(6/11)(PC)(√[(1+cos(C))/2])
Solving for PC, we get:
PC = 55/26
Substituting this value into the equation for AP, we get:
AP = 30/143
Step 3: Find BP
Using the Angle Bisector Theorem again, we have:
BP/PC = AB/AC
BP/(55/26) = 6/1
BP = 330/55
BP = 6
Therefore, BP = 6.
Final Answer: BP = 6.
Community Answer
In triangle abc AB = 6 ,BC = 5 ,CA= 4 and AP bisects angle A.if p lies...
This method can be long but you can solve it using coordinate geometry.
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In triangle abc AB = 6 ,BC = 5 ,CA= 4 and AP bisects angle A.if p lies on BC then BP equal to?
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In triangle abc AB = 6 ,BC = 5 ,CA= 4 and AP bisects angle A.if p lies on BC then BP equal to? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In triangle abc AB = 6 ,BC = 5 ,CA= 4 and AP bisects angle A.if p lies on BC then BP equal to? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In triangle abc AB = 6 ,BC = 5 ,CA= 4 and AP bisects angle A.if p lies on BC then BP equal to?.
In triangle abc AB = 6 ,BC = 5 ,CA= 4 and AP bisects angle A.if p lies on BC then BP equal to? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In triangle abc AB = 6 ,BC = 5 ,CA= 4 and AP bisects angle A.if p lies on BC then BP equal to? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In triangle abc AB = 6 ,BC = 5 ,CA= 4 and AP bisects angle A.if p lies on BC then BP equal to?.
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